It is often asked why are there so many different weather forecast models. Why is there not just one? How is it that their output can be so different? Which one is the best?
The three primary used synoptic forecast models are the North American Mesoscale Model or NAM (formally ETA), the Global Forecast System or GFS (formally AVN and MRF), and the long standing Nested Grid Model or NGM. There are also other models such as the RUC, Canadian Model, European Model. There are also many variants of these models and mesoscale models.
Think of a forecast model as a set of equations that are initialized and then solved through time. The quality of the initialization is going to depend on the input (weather data) and how realistic the equations are. Weather data is imperfect. First, it is impossible to know weather information at every point. The weather information is spread out by many miles and often hundreds of miles between each other (i.e. weather balloon launch points). Thus, there are huge gaps of weather information. Second, weather data requires that a measurement take place. Any sensor that is experiencing error, even very minor error, will contaminate the data that goes into the forecast model. There are several additional reasons for imperfect weather data such as unrepresentativeness error and the missing of mesoscale processes.
Not all the models have the same data input and each model has a different mathematical way that the equations are solved. There are also differences in resolution, display of output and how physical processes are integrated into the model. A variety of model differences and limitations are given on link below:
The model that does the best will depend on the particular weather situation. Some models will do well in certain weather situations but poor in others. Some models will do better in certain geographic regions but poor in others (even with the same type of weather event). It is a good idea to see how each model performs in certain weather situations for your particular geographic region. When the models have similar solutions that is an indication they may all be doing well for that particular weather situation.
In conclusion, differences in how the math is set up and the amount / quality of the weather data ingested into the model accounts for the differences in the models. The model that is the best will depend of the weather situation and geographic region.